Theorem subex-P7.VR 1043

Description: Substitution Law for '∃𝑥' (variable restriction). †

'𝑥' cannot occur in '𝛾'.

Hypothesis

Ref	Expression
subex-P7.VR.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
subex-P7.VR	⊢ (𝛾 → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Distinct variable group:   𝛾,𝑥

Proof of Theorem subex-P7.VR

Step	Hyp	Ref	Expression
1		ndnfrv-P7.1 826	. 2 ⊢ Ⅎ𝑥𝛾
2		subex-P7.VR.1	. 2 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
3	1, 2	subex-P7 1042	1 ⊢ (𝛾 → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L11 28  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by: (None)